1. Higher Maths Compound Angles Strategies Click to start
Higher Maths
Strategies
Compound Angles
Click to start

2. Compound Angles The following questions are on
Maths4Scotland Higher
The following questions are on
Compound Angles
Non-calculator questions will be indicated
You will need a pencil, paper, ruler and rubber.
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3. This presentation is split into two parts
Maths4Scotland Higher
This presentation is split into two parts
Using Compound angle formula for
Exact values
Solving equations
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4. a) Find the exact values of: i) sin (2p) ii) cos (2p)
Maths4Scotland Higher
A is the point (8, 4). The line OA is inclined at an angle p radians to the x-axis
a) Find the exact values of: i) sin (2p) ii) cos (2p)
The line OB is inclined at an angle 2p radians to the x-axis.
b) Write down the exact value of the gradient of OB. 8 4 p
Draw triangle
Pythagoras
Write down values for cos p and sin p
Expand sin (2p)
Expand cos (2p)
Hint
Use m = tan (2p)
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5. In triangle ABC show that the exact value of
Maths4Scotland Higher
In triangle ABC show that the exact value of
Use Pythagoras
Write down values for
sin a, cos a, sin b, cos b
Expand sin (a + b)
Substitute values
Hint
Simplify
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6. cos x and sin x Maths4Scotland Higher
Using triangle PQR, as shown, find the
exact value of cos 2x
Use Pythagoras
Write down values for
cos x and sin x
Expand cos 2x
Substitute values
Hint
Simplify
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7. On the co-ordinate diagram shown, A is the point (6, 8) and
Maths4Scotland Higher
On the co-ordinate diagram shown, A is the point (6, 8) and
B is the point (12, -5). Angle AOC = p and angle COB = q
Find the exact value of sin (p + q). 10 13 6 8 5 12
Mark up triangles
Use Pythagoras
Write down values for
sin p, cos p, sin q, cos q
Expand sin (p + q)
Substitute values
Hint
Simplify
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8. A and B are acute angles such that and . Find the exact value of
Maths4Scotland Higher
A and B are acute angles such that and
Find the exact value of
a) b) c) 4 3 A 12 5 B 5 13
Draw triangles
Use Pythagoras
Hypotenuses are 5 and 13 respectively
Write down sin A, cos A, sin B, cos B
Expand sin 2A
Expand cos 2A
Expand sin (2A + B)
Hint
Substitute
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9. If x° is an acute angle such that show that the exact value of
Maths4Scotland Higher
If x° is an acute angle such that
show that the exact value of 3 4 x 5
Draw triangle
Use Pythagoras
Hypotenuse is 5
Write down sin x and cos x
Expand sin (x + 30)
Substitute
Simplify
Hint
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Table of exact values

10. The diagram shows two right angled triangles
Maths4Scotland Higher
The diagram shows two right angled triangles
ABD and BCD with AB = 7 cm, BC = 4 cm and CD = 3 cm.
Angle DBC = x° and angle ABD is y°.
Show that the exact value of 5
Use Pythagoras
Write down
sin x, cos x, sin y, cos y.
Expand cos (x + y)
Substitute
Hint
Simplify
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11. The framework of a child’s swing has dimensions
Maths4Scotland Higher
The framework of a child’s swing has dimensions
as shown in the diagram. Find the exact value of sin x°
Draw triangle
Draw in perpendicular
Use Pythagoras 3 4 x 2 h
Use fact that sin x = sin ( ½ x + ½ x)
Write down sin ½ x and cos ½ x
Expand sin ( ½ x + ½ x)
Substitute
Hint
Simplify
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Table of exact values

12. cos a and sin a Maths4Scotland Higher Given that
find the exact value of 3 a
Draw triangle
Use Pythagoras
Write down values for
cos a and sin a
Expand sin 2a
Substitute values
Hint
Simplify
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13. Find algebraically the exact value of
Maths4Scotland Higher
Find algebraically the exact value of
Expand sin (q +120)
Expand cos (q +150)
Use table of exact values
Combine and substitute
Simplify
Hint
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Table of exact values

14. cos q and sin q Maths4Scotland Higher If find the exact value of a) b)5 q 4 3
Draw triangle
Use Pythagoras
Opposite side = 3
Write down values for
cos q and sin q
Expand sin 2q
Expand sin 4q (4q = 2q + 2q)
Expand cos 2q
Hint
Find sin 4q
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15. Show that the exact value of
Maths4Scotland Higher
For acute angles P and Q
Show that the exact value of 12 13 P 5 3 Q 5 4
Draw triangles
Use Pythagoras
Adjacent sides are 5 and 4 respectively
Write down sin P, cos P, sin Q, cos Q
Expand sin (P + Q)
Substitute
Hint
Simplify
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16. You have completed all 12 questions in this section
Maths4Scotland Higher
You have completed all 12 questions in this section
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17. Using Compound angle formula for
Maths4Scotland Higher
Using Compound angle formula for
Solving Equations
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18. Solve the equation for 0 ≤ x ≤  correct to 2 decimal places
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤  correct to 2 decimal places
Replace cos 2x with
Determine quadrants
Substitute A S C T
Simplify
Factorise
Hence
Discard
Hint
Find acute x
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19. Maths4Scotland Higher Equation Determine quadrants
The diagram shows the graph of a cosine function from 0 to .
a) State the equation of the graph.
b) The line with equation y = -3 intersects this graph
at points A and B. Find the co-ordinates of B.
Equation
Determine quadrants
Solve simultaneously A S C T
Rearrange
Check range
Find acute 2x
Hint
Deduce 2x
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Table of exact values

20. a) Find expressions for: i) f(g(x)) ii) g(f(x))
Maths4Scotland Higher
Functions f and g are defined on suitable domains by f(x) = sin (x) and g(x) = 2x
a) Find expressions for:
i) f(g(x)) ii) g(f(x))
b) Solve 2 f(g(x)) = g(f(x)) for 0  x  360°
Determine x
1st expression
2nd expression A S C T
Form equation
Determine
quadrants
Replace sin 2x
Rearrange
Common factor
Hint
Hence
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Table of exact values

21. Functions are defined on a suitable set of real numbers
Maths4Scotland Higher
Functions are defined on a suitable set of real numbers
Find expressions for i) f(h(x)) ii) g(h(x))
i) Show that ii) Find a similar expression for g(h(x))
iii) Hence solve the equation
1st expression
Simplifies to
2nd expression
Rearrange:
acute x
Simplify 1st expr. A S C T
Use exact values
Determine
quadrants
Similarly for 2nd expr.
Hint
Form Eqn.
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Table of exact values

22. a) Solve the equation sin 2x - cos x = 0 in the interval 0  x  180°
Maths4Scotland Higher
a) Solve the equation sin 2x - cos x = 0 in the interval 0  x  180°
b) The diagram shows parts of two trigonometric graphs,
y = sin 2x and y = cos x. Use your solutions in (a) to
write down the co-ordinates of the point P.
Replace sin 2x
Solutions for where graphs cross
Common factor
Hence
By inspection (P)
Determine x
Find y value A S C T
Coords, P
Determine quadrants
for sin x
Hint
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Table of exact values

23. Solutions are: x= 60°, 132°, 228° and 300°
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T A S C T
Factorise
Hence
Find acute x
Hint
Solutions are: x= 60°, 132°, 228° and 300°
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Table of exact values

24. Solutions are: Maths4Scotland Higher Solve the equation for 0 ≤ x ≤ 2
Rearrange
Find acute x
Note range A S C T
Determine quadrants
Solutions are:
Hint
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Table of exact values

25. Equal roots for cos q Maths4Scotland Higher
a) Write the equation cos 2q + 8 cos q + 9 = 0 in terms of cos q
and show that for cos q it has equal roots.
b) Show that there are no real roots for q
Replace cos 2q with
Try to solve:
Rearrange
Divide by 2
No solution
Hence there are no real solutions for q
Factorise
Equal roots for cos q
Deduction
Hint
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26. x = 0°, 120°, 240°, 360° Maths4Scotland Higher
Solve algebraically, the equation sin 2x + sin x = 0, 0  x  360
Replace sin 2x
Determine quadrants
for cos x
Common factor A S C T
Hence
Determine x
Hint
x = 0°, 120°, 240°, 360°
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Table of exact values

27. Find the exact solutions of 4sin2 x = 1, 0  x  2
Maths4Scotland Higher
Find the exact solutions of 4sin2 x = 1, 0  x  2
Rearrange
Take square roots
Find acute x
Determine quadrants for sin x
+ and – from the square root requires all 4 quadrants A S C T
Hint
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Table of exact values

28. Solutions are: x= 60°, 180° and 300°
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T
Factorise
Hence
Find acute x
Hint
Solutions are: x= 60°, 180° and 300°
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Table of exact values

29. Solutions are: x= 60° and 300°
Maths4Scotland Higher
Solve algebraically, the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T
Factorise
Hence
Find acute x
Discard above
Hint
Solutions are: x= 60° and 300°
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Table of exact values

30. You have completed all 12 questions in this presentation
Maths4Scotland Higher
You have completed all 12 questions in this presentation
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31. 30° 45° 60° sin cos tan 1 Table of exact values Maths4Scotland Higher
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